% This file was automatically generated % from the master list in TarskiTheorems.php. % Tarski-Szmielew's axiom system is used. % T is Tarski's B, non-strict betweenness. % E is equidistance. % Names for the axioms follow the book SST % by Schwabhäuser, Szmielew, and Tarski. % This file attempts to prove Satz6.16b. set(hyper_res). set(para_into). set(para_from). set(binary_res). set(ur_res). set(order_history). assign(max_seconds,3600). assign(max_mem,2000000). clear(print_kept). set(input_sos_first). set(back_sub). assign(bsub_hint_wt,-1). set(keep_hint_subsumers). assign(max_weight,16). assign(max_distinct_vars,4). assign(pick_given_ratio,4). assign(max_proofs,1). list(usable). % Following is axiom A1 E(x,y,y,x). % Following is axiom A2 -E(x,y,z,v) | -E(x,y,z2,v2) | E(z,v,z2,v2). % Following is axiom A3 -E(x,y,z,z) | x=y. % Following is axiom A4 T(x,y,ext(x,y,w,v)). E(y,ext(x,y,w,v),w,v). % Following is axiom A5 -E(x,y,x1,y1) | -E(y,z,y1,z1) | -E(x,v,x1,v1) | -E(y,v,y1,v1) | -T(x,y,z) | -T(x1,y1,z1) | x=y | E(z,v,z1,v1). % Following is axiom A6 -T(x,y,x) | x=y. % Following is axiom A7 -T(xa,xp,xc) | -T(xb,xq,xc) | T(xp,ip(xa,xp,xc,xb,xq),xb). -T(xa,xp,xc) | -T(xb,xq,xc) | T(xq,ip(xa,xp,xc,xb,xq),xa). % Following is axiom A8 -T(alpha,beta,gamma). -T(beta,gamma,alpha). -T(gamma,alpha,beta). % Following is Satz2.1 E(xa,xb,xa,xb). % Following is Satz2.2 -E(xa,xb,xc,xd) | E(xc,xd,xa,xb). % Following is Satz2.3 -E(xa,xb,xc,xd) | -E(xc,xd,xe,xf) | E(xa,xb,xe,xf). % Following is Satz2.4 -E(xa,xb,xc,xd) | E(xb,xa,xc,xd). % Following is Satz2.5 -E(xa,xb,xc,xd) | E(xa,xb,xd,xc). % Following is Satz2.8 E(xa,xa,xb,xb). % Following is Satz2.11 -T(xa,xb,xc) | -T(xa1,xb1,xc1) | -E(xa,xb,xa1,xb1) | -E(xb,xc,xb1,xc1) | E(xa,xc,xa1,xc1). % Following is Satz2.12 xq = xa | -T(xq,xa,xd) | -E(xa,xd,xb,xc) | xd = ext(xq,xa,xb,xc). % Following is Satz2.13 -E(xb,xc,xa,xa) | xb=xc. % Following is Satz2.14 -E(xa,xb,xc,xd) | E(xb,xa,xd,xc). % Following is Satz2.15 -T(xa,xb,xc) | -T(xA,xB,xC) | -E(xa,xb,xB,xC)| -E(xb,xc,xA,xB) | E(xa,xc,xA,xC). % Following is Satz3.1 T(xa,xb,xb). % Following is Satz3.2 -T(xa,xb,xc) | T(xc,xb,xa). % Following is Satz3.3 T(xa1,xa1,xb1). % Following is Satz3.4 -T(xa,xb,xc) | -T(xb,xa,xc) | xa = xb. % Following is Satz3.5a -T(xa,xb,xd) | -T(xb,xc,xd) | T(xa,xb,xc). % Following is Satz3.6a -T(xa,xb,xc) | -T(xa,xc,xd) | T(xb,xc,xd). % Following is Satz3.7a -T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xc,xd). % Following is Satz3.5b -T(xa,xb,xd) | -T(xb,xc,xd) | T(xa,xc,xd). % Following is Satz3.6b -T(xa,xb,xc) | -T(xa,xc,xd) | T(xa,xb,xd). % Following is Satz3.7b -T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xb,xd). % Following is Satz3.13a alpha != beta. % Following is Satz3.13b beta != gamma. % Following is Satz3.13a alpha != gamma. % Following is Satz3.14a T(xa,xb,ext(xa,xb,alpha,gamma)). % Following is Satz3.14b xb != ext(xa,xb,alpha,gamma). % Following is Satz3.17 -T(xa,xb,xc) | -T(xa1,xb1,xc) | -T(xa,xp,xa1) | T(xp,crossbar(xa,xb,xc,xa1,xb1,xp),xc). -T(xa,xb,xc) | -T(xa1,xb1,xc) | -T(xa,xp,xa1) | T(xb,crossbar(xa,xb,xc,xa1,xb1,xp),xb1). % Following is Satz4.2 -IFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xd,xb1,xd1). % Following is Satz4.3 -T(xa,xb,xc) | -T(xa1,xb1,xc1) | -E(xa,xc,xa1,xc1) | -E(xb,xc,xb1,xc1) | E(xa,xb,xa1,xb1). % Following is Satz4.5 -T(xa,xb,xc) | -E(xa,xc,xa1,xc1) | T(xa1,insert(xa,xb,xa1,xc1),xc1). -T(xa,xb,xc) | -E(xa,xc,xa1,xc1) | E3(xa,xb,xc,xa1,insert(xa,xb,xa1,xc1),xc1). % Following is Satz4.6 -T(xa,xb,xc) | -E3(xa,xb,xc,xa1,xb1,xc1) | T(xa1,xb1,xc1). % Following is Satz4.11a -Col(xa,xb,xc) | Col(xb,xc,xa). % Following is Satz4.11b -Col(xa,xb,xc) | Col(xc,xa,xb). % Following is Satz4.11c -Col(xa,xb,xc) | Col(xc,xb,xa). % Following is Satz4.11d -Col(xa,xb,xc) | Col(xb,xa,xc). % Following is Satz4.11e -Col(xa,xb,xc) | Col(xa,xc,xb). % Following is Satz4.12 Col(xa,xa,xb). % Following is Satz4.13 -Col(xa,xb,xc) | - E3(xa,xb,xc,xa1,xb1,xc1) | Col(xa1,xb1,xc1). % Following is Satz4.14 -Col(xa,xb,xc) | -E(xa,xb,xa1,xb1) | E3(xa,xb,xc,xa1,xb1,insert5(xa,xb,xc,xa1,xb1)). % Following is Satz4.16 -FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | xa = xb | E(xc,xd,xc1,xd1). % Following is Satz4.17 xa = xb | -Col(xa,xb,xc) | -E(xa,xp,xa,xq) | -E(xb,xp,xb,xq) |E(xc,xp,xc,xq). % Following is Satz4.18 xa = xb | -Col(xa,xb,xc) | -E(xa,xc,xa,xc1) | -E(xb,xc,xb,xc1) | xc = xc1. % Following is Satz4.19 -T(xa,xc,xb) | -E(xa,xc,xa,xc1) | -E(xb,xc,xb,xc1) | xc = xc1. % Following is Satz5.1 xa = xb | -T(xa,xb,xc) | -T(xa,xb,xd) | T(xa,xc,xd) | T(xa,xd,xc). % Following is Satz5.2 xa = xb | -T(xa,xb,xc) | -T(xa,xb,xd)| T(xb,xc,xd) | T(xb,xd,xc). % Following is Satz5.3 -T(xa,xb,xd) | -T(xa,xc,xd) | T(xa,xb,xc) | T(xa,xc,xb). % Following is Satz5.5a -le(xa,xb,xc,xd) | T(xa,xb,ins(xc,xd,xa,xb)). -le(xa,xb,xc,xd) | E(xa,ins(xc,xd,xa,xb),xc,xd). -le(xa,xb,xc,xd) | ins(xc,xd,xa,xb) = ext(xa,xb,insert(xa,xb,xc,xd),xd). % Following is Satz5.5b le(xa,xb,xc,xd) | -T(xa,xb,xe) | -E(xa,xe,xc,xd). % Following is Satz5.6 -le(xa,xb,xc,xd) | -E(xa,xb,xa1,xb1) | - E(xc,xd,xc1,xd1) | le(xa1,xb1,xc1,xd1). % Following is Satz5.7 le(xa,xb,xa,xb). % Following is Satz5.8 -le(xa,xb,xc,xd) | - le(xc,xd,xe,xf) | le(xa,xb,xe,xf). % Following is Satz5.9 -le(xa,xb,xc,xd) | -le(xc,xd,xa,xb) | E(xa,xb,xc,xd). % Following is Satz5.10 le(xa,xb,xc,xd) | le(xc,xd,xa,xb). % Following is Satz5.11 le(xa,xa,xc,xd). % Following is Satz5.12a1 -Col(xa,xb,xc) | -T(xa,xb,xc) | le(xa,xb,xa,xc). % Following is Satz5.12a2 -Col(xa,xb,xc) | -T(xa,xb,xc) | le(xb,xc,xa,xc). % Following is NarbouxLemma1 -T(xa,xb,xc) | -E(xa,xc,xa,xb) | xc = xb. % Following is Satz5.12b -Col(xa,xb,xc) | T(xa,xb,xc) | -le(xa,xb,xa,xc) | -le(xb,xc,xa,xc). % Following is Satz6.2a xa = xp | xb = xp | xc = xp | -T(xa,xp,xc) | -T(xb,xp,xc) | sameside(xa,xp,xb). % Following is Satz6.2b xa = xp | xb = xp | xc = xp | -T(xa,xp,xc) | T(xb,xp,xc) | -sameside(xa,xp,xb). % Following is Satz6.3a -sameside(xa,xp,xb) | xa != xp. -sameside(xa,xp,xb) | xb != xp. -sameside(xa,xp,xb) | c63(xa,xp,xb) != xp. -sameside(xa,xp,xb) | T(xa,xp,c63(xa,xp,xb)). -sameside(xa,xp,xb) | T(xb,xp,c63(xa,xp,xb)). % Following is Satz6.3b sameside(xa,xp,xb) | xa=xp | xb = xp | xc = xp | -T(xa,xp,xc) | -T(xb,xp,xc). % Following is Satz6.4a -sameside(xa,xp,xb) | Col(xa,xp,xb). -sameside(xa,xp,xb) | -T(xa,xp,xb). % Following is Satz6.4b sameside(xa,xp,xb) | -Col(xa,xp,xb) | T(xa,xp,xb). % Following is Satz6.5 xa = xp | sameside(xa,xp,xa). % Following is Satz6.6 -sameside(xa,xp,xb) | sameside(xb,xp,xa). % Following is Satz6.7 -sameside(xa,xp,xb) | -sameside(xb,xp,xc) | sameside(xa,xp,xc). % Following is Satz6.11a xr = xa | xb = xc | sameside(insert(xb,xc,xa,xr),xa,xr). xr = xa | xb = xc | E(xa,insert(xb,xc,xa,xr),xb,xc). % Following is Satz6.11b xr = xa | xb = xc | -sameside(xp,xa,xr) | -E(xa,xp,xb,xc) | -sameside(xq,xa,xr) | -E(xa,xq,xb,xc) | xp=xq. % Following is Satz6.13a -sameside(xa,xp,xb) | -le(xp,xa,xp,xb) | T(xp,xa,xb). % Following is Satz6.13b -sameside(xa,xp,xb) | le(xp,xa,xp,xb) | -T(xp,xa,xb). % Following is Satz6.15a xp = xq | xp = xr | -T(xq,xp,xr) | -Col(xa,xp,xq) | xa = xp | sameside(xa,xp,xq) | sameside(xa,xp,xr). % Following is Satz6.15b xp = xq | xp = xr | -T(xq,xp,xr) | -sameside(xa,xp,xq) | Col(xa,xp,xq). % Following is Satz6.15c xp = xq | xp = xr | -T(xq,xp,xr) | -sameside(xa,xp,xr) | Col(xa,xp,xq). % Following is Satz6.15d xp = xq | xp = xr | -T(xq,xp,xr) | xa != xp | Col(xa,xp,xq). % Following is Satz6.16a xa=xb | -T(xc,xa,xb) | -T(xd,xa,xb) | T(xd,xc,xb) | T(xc,xd,xb). % Following defines the function insert insert(xa,xb,xa1,xc1) = ext(ext(xc1,xa1,alpha,gamma),xa1,xa,xb). % Following is Definition Defn2.10 -AFS(xa,xb,xc,xd,za,zb,zc,zd) | T(xa,xb,xc). -AFS(xa,xb,xc,xd,za,zb,zc,zd) | T(za,zb,zc). -AFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xb,za,zb). -AFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xb,xc,zb,zc). -AFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xd,za,zd). -AFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xb,xd,zb,zd). -T(xa,xb,xc) | -T(za,zb,zc) | -E(xa,xb,za,zb) | -E(xb,xc,zb,zc) | -E(xa,xd,za,zd) | -E(xb,xd,zb,zd)| AFS(xa,xb,xc,xd,za,zb,zc,zd). % Following is Definition Defn4.1 -IFS(xa,xb,xc,xd,za,zb,zc,zd) | T(xa,xb,xc). -IFS(xa,xb,xc,xd,za,zb,zc,zd) | T(za,zb,zc). -IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xc,za,zc). -IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xb,xc,zb,zc). -IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xd,za,zd). -IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xc,xd,zc,zd). -T(xa,xb,xc) | -T(za,zb,zc) | -E(xa,xc,za,zc) | -E(xb,xc,zb,zc) | -E(xa,xd,za,zd) | -E(xc,xd,zc,zd)| IFS(xa,xb,xc,xd,za,zb,zc,zd). % Following is Definition Defn4.4 -E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa1,xa2,xb1,xb2). -E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa1,xa3,xb1,xb3). -E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa2,xa3,xb2,xb3). -E(xa1,xa2,xb1,xb2) | -E(xa1,xa3,xb1,xb3) | -E(xa2,xa3,xb2,xb3) | E3(xa1,xa2,xa3,xb1,xb2,xb3). % Following is Definition Defn4.10 -Col(xa,xb,xc) | T(xa,xb,xc) | T(xb,xc,xa) | T(xc,xa,xb). Col(xa,xb,xc) | -T(xa,xb,xc). Col(xa,xb,xc) | -T(xb,xc,xa). Col(xa,xb,xc) | -T(xc,xa,xb). % Following is Definition Defn4.15 -FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | Col(xa,xb,xc). -FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E3(xa,xb,xc,xa1,xb1,xc1). -FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xa,xd,xa1,xd1). -FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xd,xb1,xd1). -Col(xa,xb,xc) | -E3(xa,xb,xc,xa1,xb1,xc1) | -E(xa,xd,xa1,xd1) | -E(xb,xd,xb1,xd1) | FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1). % Following is Definition Defn5.4 -le(xa,xb,xc,xd) | T(xc,insert(xa,xb,xc,xd),xd). -le(xa,xb,xc,xd) | E(xa,xb,xc,insert(xa,xb,xc,xd)). -T(xc,y,xd) | -E(xa,xb,xc,y) | le(xa,xb,xc,xd). % Following is Definition Defn6.1 -sameside(xa,xp,xb) | xa != xp. -sameside(xa,xp,xb) | xb != xp. -sameside(xa,xp,xb) | T(xp,xa,xb) | T(xp,xb,xa). -T(xp,xa,xb) | xb=xp | xp=xa | sameside(xa,xp,xb). -T(xp,xb,xa) | xb=xp | xp=xa | sameside(xa,xp,xb). end_of_list. list(sos). % Following is the negated form of Satz6.16b p != q. cs != p. Col(p,q,cs). Col(p,q,r). -Col(p,cs,r). end_of_list.